Asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three. II

TL;DR

This paper proves the asymptotic stability of certain standing waves in a 3D nonlinear Schrödinger equation with a concentrated nonlinearity, showing relaxation to equilibrium without the Fermi Golden Rule in a subcritical regime.

Contribution

It establishes asymptotic stability for a class of standing waves in a 3D NLS with concentrated nonlinearity, extending stability analysis to a subcritical nonlinear regime without relying on the Fermi Golden Rule.

Findings

Existence of orbitally stable standing waves for certain nonlinearities.

Asymptotic relaxation towards standing states in a specific nonlinear range.

Stability results in an $L^2$-subcritical setting.

Abstract

We investigate the asymptotic stability of standing waves for a model of Schr\"odinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $\delta$) interaction with strength $\alpha$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $\Phi_\omega$, and the linearization around them admits two imaginary eigenvalues which in principle could correspond to non decaying states, so preventing asymptotic relaxation towards an equilibrium orbit. Without using the Fermi Golden Rule we prove that, in the range $(\frac{1}{\sqrt 2},\sigma^*)$ for a certain $\sigma^* \in (\frac{1}{\sqrt{2}}, \frac{\sqrt{3} +1}{2 \sqrt{2}}]$, the dynamics near the orbit of a standing wave asymptotically relaxes towards a standing state. Contrarily to the main results in the field, the admitted nonlinearity is $L^2$-subcritical.